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June 29, 2013
Geometry 2-3 Conditional Statements
A conditional statement is one that can be written in if-then
form. There are two parts to these statements: the
hypothesis (the "if" part) and the conclusion (the "then" part).
Identify the hypothesis and the conclusion for each statement.
Sometimes a conditional statement is written in such a way
that the words if and then are not used. In these cases, the
way to determine the hypothesis and the conclusion is to
remember that the conclusion ALWAYS depends on the
hypothesis.
If a polygon has six sides, then it is a hexagon.
q: it is a hexagon
p: a polygon has six sides
Another performance will be scheduled if the first one is sold out.
q: another performance will
p: the first one is sold out
be scheduled
Identify the hypothesis and conclusion of each statement.
Write the statement in conditional form.
The sum of the measures of two supplementary angles is 180.
p: two supp angles
q: sum of measures is 180
If two angles are supplementary, then the sum of their
measures is 180.
Just as the hypothesis and the conclusion have truth
values, conditional statements also have truth values.
We can write a conditional statement as "If p, then q." In
symbols, this is written p q. We read this as "p implies q."
Something to remember when determining the truth value of
a conditional statement: If the truth value cannot be
determined to be false, then it is considered to be true.
Determine the truth value of the conditional statement. If
false, give a counterexample.
If next month is August, then this month is July.
If angle A is acute, then its measure is 35.
TRUE
FALSE - Suppose
angle A is 40.
The original conditional and its contrapositive have the same
truth value. The converse and the inverse have the same
truth value.
Write the converse, inverse, and contrapositive of the
conditional statement. Determine the truth value of each.
If a polygon is a rectangle, then it has 4 sides.
Converse: If a polygon has 4 sides, then it is a rectangle. FALSE
Inverse: If a polygon is not a rectangle, then it does not have 4
sides. FALSE
Contrapositive: If a polygon does not have 4 sides, then it is not a
rectangle. TRUE
There are three related statements based on the conditional.
The converse is formed by exchanging the original
hypothesis and conclusion. If q, then p. q p
The inverse is formed by negating the original hypothesis
and the conclusion. If ~p, then ~q. ~p ~q
The contrapositive is formed by both exchanging and negating the
original hypothesis and conculsion. If ~q, then ~p. ~q ~p